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| Mirrors > Home > MPE Home > Th. List > nfcnv | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) |
| Ref | Expression |
|---|---|
| nfcnv.1 | ⊢ Ⅎ𝑥𝐴 |
| Ref | Expression |
|---|---|
| nfcnv | ⊢ Ⅎ𝑥◡𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-cnv 5693 | . 2 ⊢ ◡𝐴 = {〈𝑦, 𝑧〉 ∣ 𝑧𝐴𝑦} | |
| 2 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥𝑧 | |
| 3 | nfcnv.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 4 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥𝑦 | |
| 5 | 2, 3, 4 | nfbr 5190 | . . 3 ⊢ Ⅎ𝑥 𝑧𝐴𝑦 |
| 6 | 5 | nfopab 5212 | . 2 ⊢ Ⅎ𝑥{〈𝑦, 𝑧〉 ∣ 𝑧𝐴𝑦} |
| 7 | 1, 6 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥◡𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: Ⅎwnfc 2890 class class class wbr 5143 {copab 5205 ◡ccnv 5684 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-cnv 5693 |
| This theorem is referenced by: nfrn 5963 nfpred 6326 nffun 6589 nff1 6802 nfsup 9491 nfinf 9522 gsumcom2 19993 ptbasfi 23589 mbfposr 25687 itg1climres 25749 funcnvmpt 32677 nfwsuc 35819 aomclem8 43073 rfcnpre1 45024 rfcnpre2 45036 smfpimcc 46823 |
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